Question

Assume that your father is now 50 years old, plans to retire in 10 years, and expects to live for 25 years after he retires - that is, until age 85. He wants his first retirement payment to have the same purchasing power at the time he retires as $55,000 has today. He wants all his subsequent retirement payments to be equal to his first retirement payment. (Do not let the retirement payments grow with inflation: Your father realizes that if inflation occurs the real value of his retirement income will decline year by year after he retires). His retirement income will begin the day he retires, 10 years from today, and he will then receive 24 additional annual payments. Inflation is expected to be 3% per year from today forward. He currently has $100,000 saved and expects to earn a return on his savings of 4% per year with annual compounding.

Required:
To the nearest dollar, how much must he save during each of the next 10 years (with equal deposits being made at the end of each year, beginning a year from today) to meet his retirement goal? (Note: Neither the amount he saves nor the amount he withdraws upon retirement is a growing annuity.) Do not round intermediate steps.

Answer 1

$87,696

Step-by-step explanation:

your father wants to get the same distribution during the whole 25 years that he is retired, but we must first determine the initial adjusted to inflation. The $55,000 that he currently earns will be equivalent to $55,000 x (1 + 3%)¹⁰ = $73,915.40 in 10 years.

Since your father wants to start collecting the distributions immediately after he retires, this is an annuity due. Using the present value of an annuity due formula, we can determine the money that he will need to have in 10 years.

PV = annual distribution x annuity factor

• annual distribution = $73,915.40
• PV annuity due factor, 25 periods, 4% = 16.24696

PV = $73,915.40 x 16.24696 = $1,200,900.55

That PV now becomes our future value that must be saved.

Since your father already has $100,000 in his account, that will turn into $100,000 x (1 + 4%)¹⁰ = $148,024.43

This means that he is $1,200,900.55 - $148,024.43 = $1,052,876.12 short.

Using the future value of an ordinary annuity formula, we can determine his annual contribution:

annual contribution = FV / annuity factor

FV = $1,052,876.12

FV annuity factor, 4%, 10 periods = 12.006

annual contribution = $1,052,876.12 / 12.006 = $87,695.83 ≈ $87,696